Transcript Document

Hypothesis Testing
After 2 hours of frustration trying to fill out an
IRS form, you are skeptical about the IRS
claim that the form takes 15 minutes on
average to complete. How would you
challenge the IRS claim?
Methods to Test a Claim
 One method would be to find a random
sample and find a confidence interval for
the average amount of time to fill out the
form, and then determine whether the
interval suggests an average different from
15 minutes.
 Another method would be to conduct an
Hypothesis Test
The Null Hypothesis
 Is denoted by H0
 It is the statement that is under investigation
or being tested. It asserts there is no change.
 For the IRS form, the null hypothesis is:
H0:  = 15 minutes
The Alternative Hypothesis
 Is denoted by H1
 This is the statement you will adopt in the
situation the evidence (data) is so strong
that you reject the null hypothesis.
 For the IRS form, the alternative
hypothesis could be:
H1:  > 15 minutes
Three Types of Tests
 Left-tailed Tests: H0:  = k; H1:  < k
 Right-tailed Tests: H0:  = k; H1:  > k
 Two-tailed Tests: H0:  = k; H1:  ≠ k
Type of Test to Use
 This depends on what you suspect. For the IRS
form, you suspected the mean was greater time
was greater than claimed, so you would lean to a
right-tailed test.
 If you suspect that the average length of time you
get from your phone battery is less than claimed,
you would use a left-tailed test.
 Or perhaps you wish to test the average chlorine
level in a pool (too high is harmful, and too low is
not sanitary) so you might test whether it is
different from the target (two tailed test).
Data to Collect
 You will collect information similar to that
done for confidence intervals.
 If the distribution is not normal, you will
need a sample size of at least 30 to test the
mean. If the population standard deviation
is known, you will use z, if it is not known,
you may use t, especially if the sample is
not very large.
Test Statistic
 If the distribution is normal (or the sample
size is larger than 30) and the standard
deviation is known. Then
x 
z/
n
P-values
 Assuming H0 is true, the probability that the
test statistic will take on values as or more
extreme than the observed test statistic
(computed from the sample data) is called
the P-value of the test.
 The smaller the P-value, the stronger the
evidence against H0.
Types of Errors
 A Type I Error occurs if we reject the null
hypothesis when it is true.
 A Type II error occurs if we fail to reject the null
hypothesis if it is false.
 A type I error is analogous to convicting an
innocent person for a crime they didn’t commit.
 A type II error is analogous to failing to convict a
guilty person.
Level of Significance
 The level of significance  is the probability of rejecting
the null hypothesis when it is true.
 A common level of significance is .05 (that means if we
reject the null hypothesis, we will be at least 95% sure that
the null hypothesis is false).
 We will reject the null hypothesis if P-value ≤ 
 If P-value > , we do not reject the null hypothesis.
(Courts do not prove people innocent, they fail to convict
them--so failing to reject the null hypothesis doesn’t mean
it is true)
Summary of Hypothesis Tests
 Determine the null and alternative hypotheses
and set the level of significance .
 Collect the data and compute the test statistic.
 Compute the P-value.
 If P-value ≤ , then reject H0. If P-value > ,
then do not reject H0.
Further Remarks
 Roughly, the P-value is measuring how rare your test
statistic is given the hypothesized mean. A small Pvalue indicates are rare event under the hypothesis.
Rather than concluding it is a rare event, the more
likely conclusion is that the hypothesis is not correct.
 The basic principle for rareness is that measurements
far away from the mean in terms of standard
deviation are rare.
9.1#12: P/E of Bank Stocks
 Is the price to earnings ratio of US bank stocks less
than the S&P 500 mean of 19? A random sample of
14 such bank stocks had a sample mean of 17.1.
Assume the P/E ratios of US bank stocks are
normally distributed with a standard deviation of 4.5.
Conduct a left tailed test at a 5% level of significance
to determine if the mean P/E of US bank stocks is
less than 19.
9.1#13: Hail Damage
 Nationally, approximately 11% of the total U.S.
wheat crop is destroyed by hail each year. A random
sample of 16 such claims in Weld County Colorado
found the an average of 12.5% of crops destroyed by
hail. Assume the percentage of crops destroyed by
hail in Weld County is normally distributed with a
standard deviation of 5.0%. Do these data indicate
that the Weld County average is different from the
national average (in either direction)? Test at =.01
Hypothesis Tests when  is
unknown
 Follow same procedures as before. If
distribution is not approximately normal,
then the sample size must be at least 30.
 Except use the t-distribution with d.f.=n-1
and the test statistic will be
x 
t  s/
n
Critical Region Method
 As with previous method for hypothesis tests,
determine H0, H1 and .
 Instead of waiting to compute P-value and
compare to , you predetermine the critical
region, that is the values of the test statistic at
which you will reject H0.
 Then compute test statistic, and if it is in the
critical region, reject H0 otherwise do not reject
H0 .